We exploit the abstraction of the audio space to define
unique audio dimensions that make up the various pieces of the
notation. These dimensions can be thought of as lines[+] determined by a combination
of the speech and non-speech dimensions described in c:afl. The
AFL states used to produce different pieces of the audio
notation are reached by ``moving'' along these dimensions. The
functions used to generate new states are monotonic in the
mathematical sense described in eq:monotonic.

We choose unique audio dimensions to map the quasi-prefix
form into audio space. The quasi-prefix representation is a
tree with attributes. We pick one audio dimension, denoted by
dim-children (see fig:children), along which to vary
the current AFL state as different levels of a tree are
rendered. We next choose dimensions orthogonal to
dim-children to cue the visual attributes as follows.
Let [tex2html_wrap5658] and [tex2html_wrap5660] denote two
speech-space dimensions that are orthogonal to
dim-children. Select three lines in the speech space,
[tex2html_wrap5662], [tex2html_wrap5664], and
[tex2html_wrap5666]. Moving forward or backward along these
three lines cues the six visual attributes.

Conventional mathematical notation has built up a strong
association between the superscript and subscript, in that we
intuitively think of them as opposites, i.e., the
superscript moves up, and the subscript moves down. AsTeR takes
advantage of this association by moving the AFL state
``forward'' along the line [tex2html_wrap5668] before rendering
superscripts and ``backward'' along this same line before
rendering subscripts. States along the line [tex2html_wrap5670]
cue left superscripts and subscripts; states along
[tex2html_wrap5672] cue accents and underbars. By our choice of
[tex2html_wrap5674] and [tex2html_wrap5676], these variations
are independent of dimension dim-children. See
fig:superscript and fig:subscript for the audio dimensions that
are currently used for cueing superscripting and
subscripting.

[thesisfigure1389]

: Audio dimension used for rendering
subtrees.

The effect of moving along the audio dimension shown in
fig:children is to produce a softer, more animated voice. As
deeper levels of nesting are entered, the change in voice
characteristic produces a sense of falling off into the
distance.

[thesisfigure1396]

: Audio dimension used for rendering
superscripts.

A change along the audio dimension shown in fig:superscript
produces a higher pitched voice. The change in the head size
keeps the voice from sounding unpleasant. The step size along
both the average-pitch and head-size dimensions are reduced.
This allows unambiguous rendering of subscripts in
superscripts. The change in AFL state in fig:subscript is the
exact opposite of the change in fig:superscript.

[thesisfigure1406]

: Audio dimension used for rendering
subscripts.

In cases where no contextual information is available, the
visual attributes appearing on a math object are rendered in
the following order:

Subscript.

Superscript.

Underbar.

Accent.

Left-subscript.

Left-superscript.

The above ordering is motivated by the fact that in traditional
mathematical notation, the subscript binds[+] the
tightest. The order in which attributes are rendered is
encapsulated in Lisp variable
*attributes-reading-order* and may be changed by a
user.

In style simple, a commonly used rendering style,
subscripts and superscripts are rendered by first moving either
backwards or forwards along the audio dimensions shown in
fig:superscript and fig:subscript. This produces extremely
concise and unambiguous renderings. Consider the following
expressions:

[equation1420]

[equation1425]

Here, a plain verbal rendering produces an unnecessarily
complicated description that makes it difficult to comprehend
the inherent structure present in the expression.

Here is an example to illustrate the benefits of an audio
notation when rendering unusual mathematical notation. In the
following, [tex2html_wrap5678] denotes addition modulo
[tex2html_wrap5680]. Given this information,

[displaymath5682]

could be spoken as ``x plus mod n y plus mod n z''. However,
if this information is unavailable, AsTeR can still produce a
rendering that can be correctly interpreted by a listener who
is aware of the fact that the

[displaymath5684]

sign can be subscripted. Further, the listener who is
familiar with

[displaymath5686]

denoting modulo arithmetic can now understand the
expression.

In style descriptive, new AFL states are used only
if necessary when rendering superscripts and subscripts.
Typically, ``x 1'' in traditional spoken math means

[displaymath5688]

. Rendering style descriptive takes advantage of
this convention to avoid using new AFL states when rendering
subscripts that are simple. Note, however, that by doing so,
rendering style descriptive does introduce ambiguity
in the renderings;

[displaymath5690]

and

[displaymath5692]

will sound the same. In our experience, we have found that
this ambiguity is not a problem when rendering mathematical
texts; few authors write